Geometry of Surfaces: A Practical Guide for Mechanical Engineers

Geometry of Surfaces: A Practical Guide for Mechanical Engineers is a comprehensive guide to applied geometry of surfaces with focus on practical applications in various areas of mechanical engineering. It is divided into three sections, covering Part Surfaces, Geometry of Contact of Part Surfaces and Mapping of the Contacting Part Surfaces. It focuses on the geometry of real part surfaces, and explores the difference between classical differential geometry of surfaces and engineering geometry of surfaces.   Geometry of Surfaces: A Practical Guide for Mechanical Engineers combines differential geometry and gearing theory and presents new developments in the elementary theory of enveloping surfaces. Written by a leading expert in the field, this book also provides the reader with the tools for solving complex engineering problems in the field of mechanical engineering.  Key features:  Presents in–depth coverage of the analysis and synthesis of the geometry of part surfaces Contains numerous practical examples of application of the theories proposed Considers new developments in the elementary theory of enveloping surfaces Proposes a novel approach for generating of conjugate part surfaces Provides a precise analytical description of the geometry of contact between two smooth regular part surfaces Geometry of Surfaces: A Practical Guide for Mechanical Engineers is an invaluable reference for researchers and practitioners in mechanical, automotive and aerospace engineering industries, and will also be ideal source of information for CAD developers and graduate students.

About the Author xiii Preface xv Acknowledgments xvii Glossary xix Notation xxi Introduction xxv Part I PART SURFACES 1 1 Geometry of a Part Surface 3 1.1 On the Analytical Description of Ideal Surfaces 3 1.2 On the Difference between Classical Differential Geometry and Engineering Geometry of Surfaces 6 1.3 On the Analytical Description of Part Surfaces 7 1.4 Boundary Surfaces for an Actual Part Surface 9 1.5 Natural Representation of a Desired Part Surface 11 1.6 Elements of Local Geometry of a Desired Part Surface 19 2 On the Possibility of Classification of Part Surfaces 27 2.1 Sculptured Part Surfaces 27 2.2 Planar Characteristic Images 33 2.3 Circular Diagrams at a Surface Point 42 2.4 One More Useful Characteristic Curve 53 Part II GEOMETRY OF CONTACT OF PART SURFACES 55 3 Early Works in the Field of Contact Geometry 57 3.1 Order of Contact 57 3.2 Contact Geometry of Part Surfaces 59 3.3 Local Relative Orientation of the Contacting Part Surfaces 59 3.4 First–Order Analysis: Common Tangent Plane 64 3.5 Second–Order Analysis 65 3.6 A Characteristic Curve Ir k ( R ) of Novel Kind 75 4 An Analytical Method Based on Second Fundamental Forms of the Contacting Part Surfaces 79 5 Indicatrix of Conformity of Two Smooth Regular Surfaces in the First Order of Tangency 83 5.1 Preliminary Remarks 83 5.2 Indicatrix of Conformity for Two Smooth Regular Part Surfaces in the First Order of Tangency 87 5.3 Directions of Extremum Degree of Conformity of Two Part Surfaces in Contact 94 5.4 Asymptotes of the Indicatrix of Conformity Cnf R ( P 1/ P 2) 97 5.5 Comparison of Capabilities of Indicatrix of Conformity Cnf R ( P 1 / P 2) and of Dupin Indicatrix of the Surface of Relative Curvature Dup ( R ) 98 5.6 Important Properties of Indicatrix of Conformity Cnf R ( P / T ) of Two Smooth Regular Part Surfaces 99 5.7 The Converse Indicatrix of Conformity of Two Regular Part Surfaces in the First Order of Tangency 99 6 Plücker Conoid: More Characteristic Curves 101 6.1 Plücker Conoid 101 6.2 On Analytical Description of Local Geometry of a Smooth Regular Part Surface 105 6.3 Relative Characteristic Curve 112 7 Feasible Kinds of Contact of Two Smooth Regular Part Surfaces in the First Order of Tangency 117 7.1 On the Possibility of Implementation of the Indicatrix of Conformity for the Purposes of Identification of the Actual Kind of Contact of Two Smooth Regular Part Surfaces 117 7.2 Impact of Accuracy of the Computation on the Parameters of the Indicatrices of Conformity Cnf R ( P 1 / P 2) 121 7.3 Classification of Possible Kinds of Contact of Two Smooth Regular Part Surfaces 122 Part III MAPPING OF THE CONTACTING PART SURFACES 131 8 R –Mapping of the Interacting Part Surfaces 133 8.1 Preliminary Remarks 133 8.2 On the Concept of R–Mapping of the Interacting Part Surfaces 134 8.3 R–mapping of a Part Surface P 1 onto Another Part Surface P 2 136 8.4 Reconstruction of the Mapped Part Surface 140 8.5 Illustrative Examples of the Calculation of the Design Parameters of the Mapped Part Surface 141 9 Generation of Enveloping Surfaces: General Consideration 145 9.1 Envelope for Successive Positions of a Moving Planar Curve 145 9.2 Envelope for Successive Positions of a Moving Surface 149 9.3 Kinematic Method for Determining Enveloping Surfaces 154 9.4 Peculiarities of Implementation of the Kinematic Method in Cases of Multi–parametric Relative Motion of the Surfaces 164 10 Generation of Enveloping Surfaces: Special Cases 167 10.1 Part Surfaces that Allow for Sliding Over Themselves 167 10.2 Reversibly Enveloping Surfaces: Introductory Remarks 169 10.3 Generation of Reversibly Enveloping Surfaces 180 10.4 On the Looseness of Two Olivier Principles 197 Conclusion 203 APPENDICES 205 Appendix A: Elements of Vector Calculus 207 A.1 Fundamental Properties of Vectors 207 A.2 Mathematical Operations over Vectors 207 Appendix B: Elements of Coordinate System Transformations 211 B.1 Coordinate System Transformation 211 B.2 Conversion of the Coordinate System Orientation 222 B.3 Transformation of Surface Fundamental Forms 223 Appendix C: Change of Surface Parameters 225 References 227 Bibliography 229 Index 233

Stephen P. Radzevich , Principal Gear Engineer, USA